Subgroups — C₂×C₈×M₁₆ #17312

Order 256.

523 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×#322	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆	C₄	1	2	yes	—
8	C₂³×C₈	C₄	1	2	yes	—
9	C₂²×C₄²	C₂²	1	1	yes	characteristic
10	#431	C₄	1	4	yes	—
11	#431	C₂²	1	4	yes	—
12	C₂×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₂²⋊C₈	C₄	1	2	yes	—
14	C₄×M₁₆	C₄	1	4	yes	—
15	C₄×M₁₆	C₂²	1	4	yes	—
16	C₂×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈	C₄	1	2	yes	—
18	C₂×C₄×C₈	C₄	1	4	yes	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	4	yes	—
21	C₂×C₄×C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
24	#322	C₂²	1	8	yes	complemented
25	C₈²	C₂²	1	8	yes	complemented
26	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂×M₁₆	C₈	1	16	yes	direct product factor
30	C₂×M₁₆	C₂×C₄	1	8	yes	—
31	C₂²×C₈	—	2	8	no	—
32	C₂²×C₈	C₂×C₄	1	2	yes	—
33	C₂²×C₈	C₂×C₄	1	8	yes	—
34	C₂²×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₈	C₂×C₄	1	4	yes	—
36	C₂²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄²	C₂×C₄	1	2	yes	—
38	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²	C₂×C₄	1	2	yes	—
41	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²	C₂×C₄	1	2	yes	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	C₂×C₄²	C₂³	1	2	yes	—
45	C₄⋊C₈	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈	C₂×C₄	1	8	yes	—
47	#97	C₂×C₄	1	8	yes	—
48	C₄×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₈	C₂×C₄	1	8	yes	—
50	C₄×C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈	C₂×C₄	1	4	yes	—
52	C₄×C₈	C₂×C₄	1	4	yes	—
53	C₄×C₈	C₂³	1	2	yes	—
54	C₄×C₈	C₂³	1	4	yes	—
55	C₄×C₈	C₂³	1	4	yes	—
56	C₄×C₈	C₂³	1	2	yes	—
57	C₂⁴	C₄²	1	1	yes	characteristic
58	C₂²×C₄	—	2	4	no	—
59	C₂²×C₄	—	2	4	no	—
60	C₂²×C₄	C₄²	1	1	yes	characteristic
61	C₂²×C₄	C₄²	1	1	yes	characteristic
62	C₂²×C₄	C₄²	1	2	yes	—
63	C₂²×C₄	C₄²	1	2	yes	—
64	C₂²×C₄	C₄²	1	2	yes	—
65	C₂²×C₄	C₄²	1	1	yes	characteristic
66	C₂²×C₄	C₂×C₈	1	8	yes	—
67	C₂²×C₄	C₂²×C₄	1	2	yes	—
68	C₂²×C₄	C₂²×C₄	1	2	yes	—
69	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄	C₂²×C₄	1	2	yes	—
71	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
73	M₁₆	C₂×C₈	1	32	yes	direct product factor
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	2	8	no	—
76	C₂×C₈	—	2	16	no	—
77	C₂×C₈	C₂×C₈	1	16	yes	complemented
78	C₂×C₈	M₁₆	1	8	yes	direct product factor
79	C₂×C₈	C₂²×C₄	1	4	yes	—
80	C₂×C₈	C₂²×C₄	1	8	yes	—
81	C₂×C₈	C₂²×C₄	1	8	yes	—
82	C₂×C₈	C₂²×C₄	1	4	yes	—
83	C₄²	C₄²	1	2	yes	—
84	C₄²	C₄²	1	2	yes	—
85	C₄²	C₄²	1	2	yes	—
86	C₄²	C₂²×C₄	1	2	yes	—
87	C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	2	yes	—
91	C₄²	C₂²×C₄	1	2	yes	—
92	C₄²	C₂⁴	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	C₄×C₈	1	4	yes	—
96	C₂³	C₂×C₄²	1	2	yes	—
97	C₂³	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	C₄×C₈	1	8	yes	—
105	C₂×C₄	C₄×C₈	1	4	yes	—
106	C₂×C₄	C₂×C₄²	1	2	yes	—
107	C₂×C₄	C₂×C₄²	1	2	yes	—
108	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄²	1	2	yes	—
110	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄	C₂²×C₈	1	4	yes	—
113	C₂×C₄	C₂²×C₈	1	8	yes	—
114	C₂×C₄	C₂×M₁₆	1	4	yes	—
115	C₂×C₄	#131	1	4	yes	—
116	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	2	8	no	—
121	C₈	C₂²×C₈	1	16	yes	complemented
122	C₈	C₂×M₁₆	1	8	yes	direct product factor
123	C₂²	—	2	2	no	—
124	C₂²	—	2	2	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×C₄×C₈	1	4	yes	—
127	C₂²	C₂×C₄×C₈	1	2	yes	—
128	C₂²	C₄×M₁₆	1	2	yes	—
129	C₂²	#405	1	2	yes	—
130	C₂²	C₂²×C₄²	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	C₂×C₄×C₈	1	2	yes	—
134	C₄	C₂×C₄×C₈	1	4	yes	—
135	C₄	C₄×M₁₆	1	2	yes	—
136	C₄	#405	1	2	yes	—
137	C₄	C₂³×C₈	1	2	yes	—
138	C₄	C₂²×M₁₆	1	2	yes	—
139	C₄	C₂×#131	1	2	yes	—
140	C₂	—	2	4	no	—
141	C₂	C₈×M₁₆	1	4	yes	direct product factor
142	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
143	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
144	C₂	C₂×#405	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₄²	C₂⁴
socle	C₂³	C₂×C₄²
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₈×M₁₆	C₂	direct (N × H)
5. C₂×#322	C₂	semidirect
6. C₂×C₈²	C₂	semidirect
24. #322	C₂²	semidirect
25. C₈²	C₂²	semidirect
29. C₂×M₁₆	C₈	direct (N × H)
73. M₁₆	C₂×C₈	direct (N × H)
77. C₂×C₈	C₂×C₈	semidirect
78. C₂×C₈	M₁₆	direct (N × H)
121. C₈	C₂²×C₈	semidirect
122. C₈	C₂×M₁₆	direct (N × H)
141. C₂	C₈×M₁₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#431	2
C₂×C₄×M₁₆	2
C₂²×C₄×C₈	2
C₈×M₁₆	2
C₂×#322	2
C₂×C₈²	2

Subgroups — C2×C8×M16 #17312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×M₁₆ #17312